The main purpose of this article is to define a quadratic analogue of the Chern character, the so-called Borel character, that identifies rational higher Grothendieck-Witt groups with a sum of rational Milnor-Witt (MW)-motivic cohomologies and rational motivic cohomologies. We also discuss the notion of ternary laws due to Walter, a quadratic analogue of formal group laws, and compute what we call the additive ternary laws, associated with MW-motivic cohomology. Finally, we provide an application of the Borel character by showing that the Milnor-Witt K-theory of a field F embeds into suitable higher Grothendieck-Witt groups of F modulo explicit torsion.

In this paper we construct a $\mathbb{Q}$-linear tannakian category $\mathsf{MEM}_{1}$ of universal mixed elliptic motives over the moduli space ${\mathcal{M}}_{1,1}$ of elliptic curves. It contains $\mathsf{MTM}$, the category of mixed Tate motives unramified over the integers. Each object of $\mathsf{MEM}_{1}$ is an object of $\mathsf{MTM}$ endowed with an action of $\text{SL}_{2}(\mathbb{Z})$ that is compatible with its structure. Universal mixed elliptic motives can be thought of as motivic local systems over ${\mathcal{M}}_{1,1}$ whose fiber over the tangential base point $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}q$ at the cusp is a mixed Tate motive. The basic structure of the tannakian fundamental group of $\mathsf{MEM}$ is determined and the lowest order terms of a set (conjecturally, a minimal generating set) of relations are deduced from computations of Brown. This set of relations includes the arithmetic relations, which describe the ‘infinitesimal Galois action’. We use the presentation to give a new and more conceptual proof of the Ihara–Takao congruences.

Let F be a field, let G = Gal(/F) be its absolute Galois group, and let R(G,k) be the representation ring of G over a suitable field k. In this preprint we construct a ring homomorphism from the mod 2 Milnor K-theory k*(F) to the graded ring grR(G,k) associated to Grothendieck's γ-filtration. We study this map in particular cases, as well as a related map involving the W-group of F, rather than G. The latter is an isomorphism in all cases considered.

Naturally this echoes the Milnor conjecture (now a theorem), which states that k*(F) is isomorphic to the mod 2 cohomology of the absolute Galois group G, and to the graded Witt ring grW(F).

The machinery developed to obtain the above results seems to have independent interest in algebraic topology. We are led to construct an analog of the classical Chern character, which does not involve complex vector bundles and Chern classes but rather real vector bundles and Stiefel-Whitney classes. Thus we show the existence of a ring homomorphism whose source is the graded ring associated to the corresponding K-theory ring KO(X) of the topological space X, again with respect to the γ-filtration, and whose target is a certain subquotient of H*(X, F2).

In order to define this subquotient, we introduce a collection of distinguished Steenrod operations. They are related to Stiefel-Whitney classes by combinatorial identities.

There is an equivalence relation on the set of smooth maps of a manifold into the stable unitary group, defined using a Chern-Simons type form, whose equivalence classes form an abelian group under ordinary block sum of matrices. This construction is functorial, and defines a differential extension of odd K-theory, fitting into natural commutative diagrams and exact sequences involving K-theory and differential forms. To prove this we obtain along the way several results concerning even and odd Chern and Chern-Simons forms.

Let d be the degree of an algebraic one-dimensional foliation $\mathcal F$ on the complex projective space ${\mathbb P}_n$ (i.e. the degree of the variety of tangencies of the foliation with a generic hyperplane). Let $\Gamma$ be an algebraic solution of degree $\delta$, and geometrical genus g. We prove, in particular, the inequality $(d-1)\delta+2-2g\geq {\mathcal B}(\Gamma)$, where ${\mathcal B}(\Gamma)$ denotes the total number of locally irreducible branches through singular points of $\Gamma$ when $\Gamma$ has singularities, and ${\mathcal B}(\Gamma)=1$ (instead of 0) when $\Gamma$ is smooth. Equivalently, when $\Gamma=\bigcap_{\lambda=1}^{n-1} S_\lambda$ is the complete intersection of n - 1 algebraic hypersurfaces $S_\lambda$, we get $(d+n-\sum_{\lambda=1}^{n-1}\delta_\lambda)\delta \geq {\mathcal B}(\Gamma)-{\mathcal E}(\Gamma)$, where $\delta_\lambda$ denotes the degree of $S_\lambda$ and ${\mathcal E}(\Gamma)=2-2g+(\sum_\lambda\delta_\lambda-(n+1))\delta$ the correction term in the genus formula. These results are also refined when $\Gamma$ is reducible.

I want to say immediately that, despite the authoritative-sounding title, I am not claiming a proof of anything like the Lichtenbaum-Quillen conjecture. My intent here is only to explain the conjecture in various special cases, but maybe from an idiosyncratic point of view.

This paper is based on the text of the Coxeter-James Lecture given at the Winter Meeting of the Canadian Mathematical Society at Montreal in December, 1992.1 would like to thank the Society for awarding me the honour of giving this talk.

Generalizing a result of Soulé we prove that for an elliptic curve E defined over an imaginary quadratic field K with complex multiplication having good ordinary reduction at the prime number p > 3 which is regular for E and the extension F of K contained in K(Ep) the dimensions of the étale K-groups are equal to the numbers predicted by Bloch and Beilinson, i.e.,